Measure Theoretical Definition
- See main article measure-preserving dynamical system.
A dynamical system may be defined formally, as a measure-preserving transformation of a sigma-algebra, the triplet (T, (X, Σ, μ), Φ) Here, T is a monoid (usually the non-negative integers), X is a set, and Σ is a topology on X, so that (X, Σ) is a σ-algebra. For every element σ in Σ, μ is its finite measure, so that the triplet (X, Σ, μ) is a probability space. A map Φ: X → X is said to be Σ-measurable if and only if, for every σ in Σ, one has Φ−1(σ) ∈ Σ. A map Φ is said to preserve the measure if and only if, for every σ in Σ, one has μ(Φ−1(σ)) = μ(σ). Combining the above, a map Φ is said to be a measure-preserving transformation of X, if it is a map from X to itself, it is Σ-measurable, and is measure-preserving. The triplet (T, (X, Σ, μ), φ), for such a Φ, is then defined to be a dynamical system.
The map Φ embodies the time evolution of the dynamical system. Thus, for discrete dynamical systems the iterates for every integer n are studied. For continuous dynamical systems, the map Φ is understood to be finite time evolution map and the construction is more complicated.
Read more about this topic: Dynamical System (definition)
Famous quotes containing the words measure, theoretical and/or definition:
“God is a concept by which we measure our pain.”
—John Lennon (1940–1980)
“The hypothesis I wish to advance is that ... the language of morality is in ... grave disorder.... What we possess, if this is true, are the fragments of a conceptual scheme, parts of which now lack those contexts from which their significance derived. We possess indeed simulacra of morality, we continue to use many of the key expressions. But we have—very largely if not entirely—lost our comprehension, both theoretical and practical, of morality.”
—Alasdair Chalmers MacIntyre (b. 1929)
“Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
—Nadine Gordimer (b. 1923)